- Rankine–Hugoniot equation
The Rankine–Hugoniot equation governs the behaviour of
shock wave s normal to the oncoming flow. It is named after physicistsWilliam John Macquorn Rankine andPierre Henri Hugoniot , French engineer, 1851-1887.The idea is to consider one-dimensional, steady flow of a fluid subject to the
Euler equations and require that mass, momentum, and energy are conserved. This gives three equations from which the two speeds, u_1 and u_2, are eliminated.It is usual to denote upstream conditions with subscript "1" and downstream conditions with subscript "2". Here, ho is density, u speed, p pressure. The symbol e means internal energy per unit mass; thus if ideal gases are considered, the
equation of state is p= ho(gamma-1)e.The following equations
:ho_1u_1= ho_2u_2,:p_1+ ho_1u_1^2=p_2+ ho_2u_2^2:e_1+frac{p_1}{ ho_1}+frac{1}{2}u_{1}^2=e_2+frac{p_2}{ ho_2}+frac{1}{2}u_{2}^2
are equivalent to the conservation of mass, momentum, and energy respectively. Note the three components to the energy flux: mechanical work, internal energy, and kinetic energy.Sometimes, these three conditions are referred to as the Rankine–Hugoniot conditions.
Eliminating the speeds gives the following relationship::2left(h_2-h_1 ight)=left(p_2-p_1 ight)cdotleft(frac{1}{ ho_1}+frac{1}{ ho_2} ight)where h=frac{p}{ ho} + e.Now if the ideal gas equation of state is used we get
:frac{p_1}{p_2}=frac{(gamma+1)-(gamma-1)frac{ ho_2}{ ho_1{(gamma+1)frac{ ho_2}{ ho_1}-(gamma-1)}
Thus, because the pressures are both positive, the density ratio is never greater than gamma+1)/(gamma-1), or about 6 for air (in which gamma is about 1.4). As the strength of the shock increases, the downstream gas becomes hotter and hotter, but the density ratio ho_2/ ho_1 approaches a finite limit of 4 for a monatomic gas (gamma = 5/3) and 6 for a diatomic gas (gamma = 1.4).
References
* Rankine, W. J. M. , " [http://gallica.bnf.fr/scripts/get_page.exe?O=55965&E=328&N=11&CD=1&F=PDF On the thermodynamic theory of waves of finite longitudinal disturbances] ", Phil. Trans. Roy. Soc. London, 160, (1870), p. 277.
* Hugoniot, H., "Propagation des Mouvements dans les Corps et spécialement dans les Gaz Parfaits", Journal de 1’Ecole Polytechnique, 57, (1887), p. 3; 58, (1889), p. 1.
* Salas, M. D. " [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20060047586_2006228914.pdf The Curious Events Leading to the Theory of Shock Waves] " Invited lecture at the 17th Shock Interaction SymposiumRome, Italy 4-8 September 2006.External links
* [http://web.ics.purdue.edu/~alexeenk/GDT/index.html Gas Dynamics Toolbox] Calculate normal shock wave parameters for mixtures of imperfect gases
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